Stochastic cusp catastrophe model and its Bayesian computations

Abstract

This paper revitalizes the investigation of the classical cusp catastrophe model in catastrophe theory and tackles the unsolved statistical inference problem concerning stochastic cusp differential equation. This model is challenging because its associated transition density hence the likelihood function is analytically intractable. We propose a novel Bayesian approach combining Hamiltonian Monte Carlo with two likelihood approximation methods, namely, Euler approximation and Hermite expansion. We validate this novel approach through a series of simulation studies. We further demonstrate potential application of this novel approach using the real USD/EUR exchange rate.

Keywords:

• Cusp catastrophe model
• stochastic differential equation
• transition density
• Hermite expansion
• Bayesian inference
• Hamiltonian Monte Carlo

2010 Mathematics Subject Classification:

• 60G25

Acknowledgements

The authors would like to thank the two anonymous reviewers, the associate editor and the editor-in-chief (Professor Jie Chen, Biostatistics and Data Science, Augusta University, USA) for their comments and suggestions, which have significantly improved the quality of this chapter.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is partially supported by the National Research Foundation of South Africa [grant number 127727) and the South African National Research Foundation (NRF) and South African Medical Research Council (SAMRC) (South African DST-NRF-SAMRC SARChI Research Chair in Biostatistics [grant number 114613].